How is VLBI delay calculated?

I’m working on a research project wherein we are trying to solve a problem very similar to VLBI delay. We have two radio receivers, and we know their locations. We also know at what time one receiver detected a signal, and the coordinates of it’s beam-pointing center. We want to determine at what time the second receiver should’ve seen the signal.

I understand in principle how VLBI delay is calculated; I’ve seen a few different formulas, such as the dot product of a vector between the receivers, and between the receiver and the source but practically speaking, how would I actually go about producing numbers?

• Yes this task needs a more definitive solution than what I gave before. Can you use Python? Or are you planning to use a spread sheet or pencil and paper, or do you need a website? I am not sure I can post a complete answer but probably someone can. But it will be helpful to them if you can mention which resources or calculation techniques you have available.
– uhoh
Jul 14, 2020 at 9:51

This is impossible to answer without clearly defining the geometry of your setup. VLBI delay and delay rate are calculated assuming the source is a long way away. This essentially means the incoming signal can be assumed to be planar waves arriving at each receiver. This significantly simplifies the maths and allows astronomers to assume the sources they are observing are projected onto a 2D celestial sphere which is a long way away.

In your specific case, if the objects is within the near field of the baseline, then the maths changes significantly and you have to consider the 3D volume the source is in. Note for the near field calculation, you need to consider the separation of the receivers, not the size of the individual receivers.

The VLBI case gets complex pretty quickly, even assuming far field. You have to consider the rotation of the earth wrt the source you are observing, as well as the full 3D spherical geometry. Add into this relativistic effects, compensation for the ionosphere and troposphere, earth tide, tectonic plate movement, you may find the VLBI case is a long way removed from what you really want.

If you really want to find some software which does the VLBI delay calculation, you could look at VTD (and links therein):

Library VTD computes a priori VLBI time delay and delay rate with precision needed for geodetic applications. Terms which contributes to delay at the level 10-13 sec for delay and 10-15 for delay rate are kept. The accuracy of VTD is determined by the accuracy of a priori values. If the a priori site positions, source coordinates, EOP, tropospheric path in zenith direction were perfect, the theoretical time delay and delay rate would be accurate to that level. Library VTD also computes the Doppler frequency shift for the Solar system objects.